Mathematics is obviously about more than computation, but mastering computational techniques broadens our horizons when we pursue research. Before we can think strategically, we must first get the technicalities out of the way. And since the likelihood of errors in a calculation increases with its length, it seems wise to do as little calculation as possible. This minimalist computational approach can be illustrated by linear recurrences and linear differential equations with constant coefficients.
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Linearity and vector spaces
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Much of the mathematics taught in the early grades is concerned, in one way or another, with linearity. Yet regular examples show that proportionality is still not properly understood either by those responsible for communicating information—typically journalists—or by those who must act on it—mainly politicians. Although easy to grasp and fundamental to all scientific progress, proportionality has far-reaching consequences. Linearity lies at the heart of the definition of vector spaces: sets of elements called vectors that are closed under linear combinations. The same closure under linear combinations appears in the definition of a linear function or operator: the image of a linear combination is the corresponding linear combination of the images. More explicitly, T(α x + β y ) = α T(x) + β T(y), where α and β are scalars in a field, a set in which the usual four operations can be performed. Let's simplify matters by working over the real numbers.
The fundamental consequence of this definition is that if we have two independent—that is, non-proportional—solutions (u, v) of the equation T(x) = 0, then every combination of the form α u + β v is also a solution of that equation:
